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Morphologic evolutionary systems
Published in Richard J. Chorley, Stanley A. Schumm, David E. Sugden, Geomorphology, 2019
Richard J. Chorley, Stanley A. Schumm, David E. Sugden
Stochastic mathematical models consist of sets of equations involving mathematical variables, parameters and constants together with one or more random components, the latter arising from random effects which may be extremely important in influencing natural processes and in producing unpredictable fluctuations in observational or experimental data. A simple type of stochastic mathematical model is the random walk process used to generate simulated branching stream networks by allowing drainage from each of a grid of squares with differing probability, depending on direction. Figure 2.16A presents a network resulting from flow which is equally probable in all four cardinal directions (i.e. P↓ = P↑ = P← = P→ = 0.25) and Figure 2.16B shows a network developed from a directionally biased model (i.e. P↓ = 0.4; P↑ = 0.1; P← = P→= 0.25). This type of model illustrates one of the major difficulties in employing mathematical simulation to explore system changes through time, namely that the model may have to be so simplified and its development proceed through such unreal and artificial steps that its only value lies not in its evolutionary representations, but in its ability to approach and exemplify some equilibrium state.
Optical Lithography Modeling
Published in Bruce W. Smith, Kazuaki Suzuki, Microlithography, 2020
Chris A. Mack, John J. Biafore, Mark D. Smith
Modeling strategies for optical lithography have historically (and successfully) relied on the continuum approximation to describe the physical world being simulated. Even though light and matter are quantized into photons and spatially distributed molecules, respectively, the calculations of aerial and latent images ignore the discrete nature of these fundamental units and instead use continuous mathematical functions and “classical” chemical rate equations. Continuum models allow the computational domain to be subdivided into ever-smaller cells, with each cell retaining the properties of the bulk. An alternative to continuum modeling and, in a real sense, a more fundamental approach is to attempt to build the quantization of light and matter directly into the models in what is called stochastic resist modeling. Stochastic models are advantageous when smaller features and shorter exposure wavelengths dominate the process of interest. Two examples of processes where stochastic modeling is routinely used are high-NA ArF and extreme ultraviolet (EUV) lithography. Stochastic modeling approaches involve the use of random variables and probability density functions to describe the expected statistical fluctuations. The lithographic metrics that we obtain through stochastic modeling are described by the first and second moments of their probability distributions—the average and the variance. Stochastic modeling can be used to investigate interesting and important phenomena that cannot be studied with continuum approaches. For example: The counting statistics of absorbed photons, molecules of resist reactants, generated acids, or other photoproductsThe effects of photon shot noise or acid (chemical) shot noiseThe effects of resist ionization and reactivity with scattering photoelectrons at EUVThe uncertainty in feature edge placement, CD, or uniformity
Variational approach for robust design and sensitivity analysis of mechatronic systems
Published in Journal of the Chinese Institute of Engineers, 2020
Hana Siala, Faïda Mhenni, Jean-Yves Choley, Maher Barkallah, Jamel LouatI, Mohamed Haddar
The variability in a mechatronic system is unavoidable. Its behavior can deviate from the required one because of variations of some parameters. The fluctuations can add up and cause a system failure. Without paying attention to the variations, the design probably fails in meeting the desired requirement (power, timing, stability, and quality) (Ghosh and Roy 2010). Parameter variation (PV) can be categorized into two classes, regarding its mathematical representation (Zerelli 2014): PV which is not time dependent. The variation but (for the time interval). The deviation is invariant in the time-scale and its initial value is unknown. It may be caused by manufacturing.PV changing over time. That means but . The initial value of uncertainty is known. In this case, we study one sample with time-varying parameters. This case can be during exploitation, for example, caused by wear or fatigue.
Frequency Control of Power System with Distributed Sources by Adaptive Type 2 Fuzzy PID Controller
Published in Electric Power Components and Systems, 2023
Srinivasan Kullapadayachi Govindaraju, Raghuraman Sivalingam, Sidhartha Panda, Preeti Ranjan Sahu, Sanjeevikumar Padmanaban
In actual systems, the parameters used to build the system are likely to be imprecise. Moreover, the settings may vary over time, influencing system performance. As a result, it is critical to analyze system performance under system parameter fluctuations. In this scenario, uncertainties in system parameters are examined to assess the adaptability and resilience of the projected control approach. The following situations are being consideration:
Robust nonfragile guaranteed cost control for uncertain T-S fuzzy Markov jump systems with mode-dependent average dwell time and input constraint
Published in International Journal of Systems Science, 2018
Takagi-Sugeno (T-S) fuzzy model (Takagi & Sugeno, 1985) proposed in 1985 provides a local linear description or approximate representation of complex nonlinear systems in terms of IF-THEN fuzzy rules. The method is closely related to any high-precision nonlinear system, and has become very practical and effective in the control problem of complex nonlinear systems. Hence, a large number of theoretical results and practical applications of T-S fuzzy systems have been reported in the past three decades (Asemani & Majd, 2015; Chen, Liu, Tong, & Lin, 2007; Huang & Yang, 2014; Lin, Wang, & Lee, 2006; Liu, Wu, He, & Yokoyama, 2010; Shen, Jiang, & Cocquempot, 2012; Wu, Shi, Su, & Lu, 2015; Xie, Yue, Zhang, & Peng, 2017; Zhang & Wang, 2015; Zhang, Jiang, & Staroswiecki, 2010; Zhao, Yin, Zhang, & Yang, 2016; Zhao & Gao, 2012). In addition, guaranteed cost control (Chang & Peng, 1972; Lu, Cheng, & Bai, 2015; Qiu, Yao, Xu, Li, & Xu, 2016; Ren & Zhang, 2012) has been rapidly developing in recent years. The main purpose is to design an appropriate controller to make the system stable with the prescribed performance. Since the inherent inaccuracies of analogue systems, running controllers often experience fluctuations, such as numerical runoff in the calculation and finite word length in digital systems. These fluctuations may result in performance degradation even with instability for systems. Therefore, designing a nonfragile controller is very important to ensure that the desired performance and powerful capabilities are achieved to tolerate admissible uncertainties of the controller gains. Recently, nonfragile guaranteed cost control problems were investigated in Refs. Zhu, Song, Zhang, and Zhong (2017), Xie and Tang (2006), Han, Wu, Lam, and Zeng (2014), Jiang, Kao, and Gao (2017), Zhang et al. (2018) and Chen and Ma (2017). In practical applications, systems may experience a sudden change in their parameters and structure. These systems can be usually modelled as Markov jump systems (MJSs) (Ji & Chizeck, 1990). In such class of stochastic systems, the jumps of the operating mode are controlled by a Markov process. Over the past decades, a great number of results for Markov jump linear systems have been investigated, including stochastic stability and stochastic stabilisation (Bolzern, Colaneri, & De Nicolao, 2014; Ji & Chizeck, 1990; Xiong, Lam, Gao, & Ho, 2005), robust filtering (Xu, Chen, & Lam, 2003), and robust control (Cao & Lam, 2000). With the rapid development of fuzzy system theory, fuzzy MJSs (FMJSs) are also proposed. Many problems of FMJSs have been investigated (Tao, Lu, Shi, Su, & Wu, 2017; Wu & Cai, 2007; Zhang, Xu, Zou, & Lu, 2011). For example, time-varying delay (Tao et al., 2017; Zhang et al., 2011) and external disturbance (Tao et al., 2017) problems in FMJSs were widely studied. A great number of results on robust control have been presented in Refs. Zhang et al. (2011) and Wu and Cai (2007) and references herein.